If the weight \(w(x)=d(x,\Omega ^c)^\beta \)  is \(p\) -admissible in the metric space \(X=(X,d,\mu )\) , that is,
the weighted measure \(w\,d\mu \)  is doubling and supports a
\((1,p)\) -Poincaré inequality, then
the weighted cases of our main results could be obtained
from the corresponding unweighted results applied
in the weighted metric space \((X,d,w\,d\mu )\) .
In particular, then for \(q=p\)  the weighted case can be directly obtained from the results in [1]}.
Many examples of admissible weights are provided by weights from the Muckenhoupt classes,
see [2]} and the remark after [2]}.
Sufficient and necessary conditions for the weight \(w(x)=d(x,\Omega ^c)^\beta \)  to be a Muckenhoupt weight
have been given in [4]} in terms of the (co)dimension of \(\Omega ^c\) .
However, we emphasize that our more general approach applies also
in the cases where \(w\)  is not admissible, and thus we obtain a unified theory for all
weighted Hardy–Sobolev inequalities.
